IN THE THEORY OF DIFFERENTIAL EQUATIONS. 541 



are two, and the rest formed on the same type. This expression remains unaltered when a 

 and x, b and y, c and z, are simultaneously interchanged : which proves the property asserted. 

 The same kind of demonstration might be applied in the case of two variables. 



21. The general method of transforming partial equations, given in the Appendix to my 

 last paper, and which I here call that of polar transformation, amounts to the investigation 

 of surfaces which possess a certain differential property by investigating their reciprocal polar 

 surfaces from the correlative polar property. We take any modular equation we please, 

 d> (x, y, x, £, t], £) =0, and make each set of co-ordinates compensatory; so that d <p = 0, 

 & ^ f <p = 0. Of this it has been shown to be a consequence that x = x (£, tj, £, tst, k) where 

 ■sr = ft, k = £. : and the same of y, z,p, q: and vice versa. We determine £, ri, t> in terms 

 of x, &c. from = 0, (p xU = 0, <f> y = 0: and in all results the letters of the two systems are 

 interchangeable. We determine p and q (when - (p T : <f> z and - <p y : <p„ followed by substi- 

 tution are not more convenient) from 



_ yfi»,i - y,i«*i > = _ *t\*,i-*ii *t\ 

 **isr,i - *,iy«i ' «*iy,i-»,iy*i' 



from which, as proved, p, a, t, (answering to r, s, t), disappear. We find r by complete dif- 

 ferentiation ofp, on the supposition that y = const., or y^d^ + y^dtj = 0, which gives 



r = i^- m ^l d £ + P" \ d * m _ yt\Pn\-y»\Pt;\ 



d.x m t J3| + «, \dt) ** |Sft, |-**I%|' 



and so on. In /(a?, y, z, p, q, r, s, t, ...) = we substitute for x, y, z, &c, and thus produce 

 F(%, *i, £, w, k, p, a, t, ...) of the same order as / = 0. If F = can be integrated, we can 

 find £, nr, k, &c. in terms of £ and tj : and we then arrive at the primitive of / = by elimi- 

 nating J and tj between w = x (£, »j), y = y (£, >?), a = z (£, ^) or else by eliminating £, >;, £, 

 p, q between the integral found and the expressions for £, r\, £, "&, k in terms of x, y, z, p, q. 



22. The notation by which eliminants (commonly called determinants) are expressed in 

 writings which treat on their properties, is judiciously made to be as full as possible : but 

 it is inconvenient for such applications as usually occur in connection with other symbols. 

 I find it convenient to denote an eliminant, say of the fourth order, by {ABCD pqr>) or 

 (ABCD ' " '"' according to the notation used for the component symbols. 



Thus, 



Of* = - / 4p> 



{AB m) -A^B^-B^-A^-B^, 



(ABC^ = A p (BC qr) t B p (CA V) + C p (AB qr) , 



{ABCD^ = A p (BCD VS) - B p {CDA qrs) + C p {DAB„ - D p (ABC^, 



(ABCDE^ = A p {BCDE vtt) + B^CDEA^ + C p {DEAB vtt) + D p (EABC^ + E p {ABCD v «, 



