1196 
MR. R. F. GWTTHER ON THE DIFFERENTIAL 
and, therefore, the coefficient of y“ in the reciprocal will be, omitting a power of Wj) 
4 « 2«4 — 
and this is seen to satisfy = 0 ( 25 ). 
On introduction of Ag, Ag, A^ to replace Yg, Yg, Y^, as a.2, rej^lace y^, and 
and omitting a power of Ag, we obtain 
A2A4 — Ag^ 
for the matrix of the reciprocal conic as a function of Ag, Ag, A^, in which form it is, 
of course, a solution of XlgF = 0.— Rewritten August 2, 1893 .] 
]. 9 . Dual relation between the o'perators and fl,- 
The methods of solution of = 0 and alike, inasmuch as each 
depends upon a method of eduction starting from the solutions of the second and 
fourth orders. The general proposition is now stated, that if u be a solution of either 
yiyf= 0 or £1.2/= 0, in which y.2, y^, &c., or, what is the same thing, a^, %, &c., are 
the arguments, and if in consequence of the substitution for these cjuantities of Yg, Yg, 
&c., or of Ag, Ag, &c., u becomes U, then U is a solution of the other corresponding 
equation, that is, of flgF = 0 or fl^F = 0. 
As far as the first three solutions of 0 ,yf = 0 are concerned, it is readily verified 
that 
Wg, W4, and u^, 
become 
Ug, dAgA^ — 5 Ag^, and U5, 
divided respectively by Ug^, — Ug®, and — Ug® and numerical factors, where the index 
of the power of Ug is the weight of the function which it affects. 
If ^ be a function of Ug, ftg, &c., of weight iv and degree on substitution it will 
become ©/Ug®, where @ has the weight and degree of 6 . Writing W and for the 
weight and degree of @/Ug® we obtain 
W = — tv, = c4 — w, 
so that 
W + = — { 2 iv — d,^) and — (2W — D.^) = w • • * (^ 5 ). 
Bearing in mind the conditions (18a) for the eduction of solutions of Hyf = 0 and 
Xlg/' = 0, it fellows that processes which educe solutions of Og/ = 0 from functions of 
the character of 6 , will educe solutions of n^F = 0 from functions of the character of 
©/Ug®, and vice versa. 
As a consequence, in making a change from point-coordinates to the correlated 
line coordinates, or vice versa, the operators Xli and fig are interchanged. Or expressed 
