cinders and burning matter, together with flames, rumbling, and earthquakes. These phenomena, which had a gradually decreas ing course, lasted through all the month of July 1874; and traces of them still remain. Stromboli also in June last had an unusual eruption, being violently agitated, and throwing out stones as far as the inha bited district which lies underneath it, showing much greater activity than in the little explosions every two or three minutes which characterize its usual action. It seems that Vesuvius also has not remained indifferent to all this; and I saw myself from its crater, as well as from that of Stromboli, a remarkable and unusual amount of thick smoke coming out at the same time that the eruption of Etna took place. XVII. On Primary Forms. By Sir JAMES COCKLE, F.R.S., Corresponding Member of the Literary and Philosophical Society of Manchester, President of the Queensland Philosophical Society, &c.* 1. COMPARING OMPARING p. 428 of Boole's 'Differential Equations' (1865), and pp. 184 and 190 of the Supplement, with a citation in Mr. Harley's recent paper "On the Theory of Differential Resolvents" (which purports to be reprinted from the Report of the British Association for 1873), it seems to me that Boole did not always use the term primary in the same sense. By primary I mean integrable, but not through Boole's reductions. By a factorial substitution I mean a change of y into Xy, where X is a function of x. By taking the criticoid of a biordinal I mean reducing its middle term to zero and dividing the equation so transformed by the coefficient of its first term. This criticoidalt transformation is effected by factorial substitution, not by change of the independent variable. By the equation of the casura, or briefly the casura, I mean an equation derived from a given equation by expunging its last term and diminishing by unity the indices of the differential coefficients. 2. The regular forms, as we may term those solved through Boole's reductions, are of two descriptions. In one factors are preserved which in the other are lost. The first description of binomial biordinal may be written {(A-2m)(A-1)~(y+2n)(−1)x2y=0, where (1) A=D+B and y=D+a. · Communicated by the Rev. Robert Harley, F.R.S. + As to this user of the term criticoidal, see my paper "On Hyperdistributives" (Phil. Mag. for April 1872) and the papers connected therewith. Let A(A—')..(A—nr)= [A]n+1. [A]+1=(Anr)[4] =A[Ar]"; so that if in (1) we replace y by its equivalent [4]m2 [+2n]" [A]" [+2n]2 then (1) will take the form Then where 1 [A] [▼+2n]n Y• Hence the solution of (1) is reduced to that of (A-1±vx)(AFуx) Y=0. But a particular integral of (2) is Y=x-(x+1)B-α-). Consequently (1) has a particular integral of the form (x+1)`R(x)=y, y, [A-2][A+2n]" {A(A−1)—▼(▼−1)x2} Y=0, where Y= But {(A-2n) (D+B)+A(D+a) x2} y=0, where A=D+ any constant. For if we replace y by then (3) reduces to [A]2+1G where R(x) is rational. 3. This form is included in ef(x)dx, where (x) is rational. So, too, is that of the solution of the first case of the second description of regular forms, viz. 1 [4]2Y=0, Hence a particular integral of (3) will be given by y=[A]"G-10. B-α-2 2 (2) G-10=x-B(x2+1) Consequently (3) will have a particular integral of the form y= (x2+1)^R (x), (3) Y, (4) where Re) is rational, and therefore of the form e(x)dx, where r) is rational. 4. The second case of the second description, viz. G[A] y=0; (6) which last is solved, with redundant constants (conf. op. cit. p. 421, et Suppl. p. 189), by y=xm(Co+C1x2 + . . + C2n−2x2n−2), (7) if we make A=D-m. (5) we substitute for y the dexter of (7). For A and A-2n And the redundancy is got rid of if in respectively annul the terms Com and Can-2+2, leaving all the n quantities Co, C2, .., C2n-2 to satisfy the remaining n-1 homogeneous conditions. Thus (5) has a particular integral, which is of the form R(x), and therefore of the form e(x)dx; R(x) being rational and entire, and (x) rational. 5. The first primary form of Boole (op. cit. p. 428) may, without loss of real generality, be written (1 + x2) dx2 + x —n2y=0, . dx {D(D-I)+[(D-2)2 —n3] x2 } y=0. . (8) (9) But this form is not truly primary when n is an integer or the half of an integer, the latter case corresponding to a quadratic resolvent. And this accords with what precedes. The complete integral of (8) or (9) is C+1(x+√x2+1)” +C_、 (x−√ x2+1)”=y. Take n a positive integer and C+1=C-1. Then y is rational and entire. Take n the half of a positive integer. Then √x2+1 dy n dx =C+1(x+√/x2 + 1)”—C_1(x−√/x2+1)"; and if C+,=1_and_C_,−−1, then √x2+11 dy will become 2n n y dx (x+√x2+1)2 +(x−√√/x2+1)2 + 2(−1)” 2n (x +√ x2 + 1)2′′ — (x−√x2+1)2n In either case* 1 dy y dx is rational, and dy in the former rational dx and entire. In both cases, therefore, y is of the form e√(x)ds, where (x) is rational. The proof may be extended to the cases of n, a negative integer, or the half thereof. In no other cases has (8) or (9) a particular integral of the form e)dr. And since (8) and (9), if soluble through Boole's reduction, must fall under one of the three forms (1), (3), or (5), the first primary form is not in general so soluble. The same may be proved for his second primary form. where z is any function of x. Suppose that 2a=- 2z dz dx mean the double where p= Then (10) is reducible to an equation with constant coefficients by a change of the independent variable. The primary (8) is thus soluble. But it may be otherwise solved. Let SU mean SUda, and let P(S&S_~) operation SSSS... continued in infinitum. Also let y=¥(x) be a particular integral of (10). Then if & be determined from the equation of the casura, viz. *The same is true of every case. the three forms de +2a=0, and from For (1), (3), and (5), together with {(A—2m)(▲−1)—a2x2}y=0, and the six other forms deduced from these by the change of x into stitute the twelve forms which the reductions of Boole solve. Both here and in the text m and n are integers. The arguments for (a), (b), and (c) respectively are analogous to those for (1), (3), and (5), and show that there is at least one particular integral of the form e(x)dx. + This method of synthetical solution may give a finite result, a series summable or otherwise, or a suggestion of the form of (x), in which a constant or constants are to be determined by substitution. For the terordinal day dx2 da +36dy+cy=0, where a, b, and c may be variable, we assume y=PS1t San S25.4(x), where (x) is a particular integral, say zero, and έ is determined by the cæsura where R(x) is rational, and therefore of the form ex)dx, where (x) is rational. 4. The second case of the second description, viz. {A (D+B) + (▲ −2n) (D+a)x2} [4]2 [4]2 reduces to or G[A]2 y=0, (1 + x2) y=0, G[A]2y=0; (6) which last is solved, with redundant constants (conf. op. cit. p. 421, et Suppl. p. 189), by y=xm(Co+C1x2+.. +С2n-2x2n-2), 2 m (7) if we make A=D―m. And the redundancy is got rid of if in (5) we substitute for y the dexter of (7). For ▲ and A-2n respectively annul the terms Coam and C2n-2x+2n, leaving all the n quantities Co, C2, Can-2 to satisfy the remaining n-1 homogeneous conditions. Thus (5) has a particular integral, which is of the form R(x), and therefore of the form es¤(x)dx; R(x) being rational and entire, and p(x) rational. 2,.., 5. The first primary form of Boole (op. cit. p. 428) may, without loss of real generality, be written dy (8) {D(D− 1) + [(D—2)2—n2]x2}y=0. . (9) But this form is not truly primary when n is an integer or the half of an integer, the latter case corresponding to a quadratic resolvent. And this accords with what precedes. The complete integral of (8) or (9) is 2n (x + √ x2+1) 22 — (x−√x2+1)2n 2n (x+√ x2 + 1)22 + (x−√ x2 + 1)22 +2(−1)” 2n (5) C+1(x+√x2+1)2+C_1(x−√x2+1)”=y. Take n a positive integer and C+1=C_1. Then y is rational and entire. Take n the half of a positive integer. Then √ x2 + 1 dy = C + 1 (x + √/x2 + 1)” −C_1(x−√/x2+1)"; n dx and if C+1=1 and C_1=-1, then √x2+11 dy will become n y dx dy In either case* is rational, and dx 1 dy y dx and entire. In both cases, therefore, y is of the form e√(x)dx, where (x) is rational. The proof may be extended to the cases of n, a negative integer, or the half thereof. In no other cases has (8) or (9) a particular integral of the form e√(x)dx. And since (8) and (9), if soluble through Boole's reduction, must fall under one of the three forms (1), (3), or (5), the first primary form is not in general so soluble. The same may be proved for his second primary form. 6. Let dzy dz +2ady +zy=0, (10) p where z is any function of x. Suppose that 2a= : - 2/2 + C √ √ %, in the former rational * The same is true of every case. the three forms dx® where p= Then (10) is reducible to an equation with con- de For (1), (3), and (5), together with (a) (b) (c) +36 dy dx2 da +2a=0, and n from η {(A-2m)(A-1)—a2x2}y=0, and the six other forms deduced from these by the change of x into con x stitute the twelve forms which the reductions of Boole solve. Both here and in the text m and n are integers. The arguments for (a), (b), and (c) respectively are analogous to those for (1), (3), and (5), and show that there is at least one particular integral of the form e√(x)dx. This method of synthetical solution may give a finite result, a series summable or otherwise, or a suggestion of the form of (x), in which a constant or constants are to be determined by substitution. For the terordinal +cy=0, d3y (a) where a, b, and c may be variable, we assume y=PS_SS 5.4(a), where ↓(≈) is a particular integral, say zero, and έ is determined by the cæsura |