AtE. A. CATLEY ON A NEW AUXILIAEY EQUATION IN 
1>T0 
17. To obtain the symmetrical functions of a, p, y, it is only necessary to remark, that 
if in the identical equation 
(1, b, c, 1)3=(^_ A)(^-BX^-C), 
we put + —\{x — b), in the place. of the equation becomes 
(1, b, c, dX%— b, 2X=(>i+a)(p^+PX%+y), 
so that we have 
2a =~b = c, 
2aP=— b^-l-4c =~lQae -\-A.hd — 
aj3y = b®— 4bc-f-8d= lQace—8ad^—8b'^e~{-4:bcd~c\ 
18. The developed expression for the equation in ip is easily found to be 
-|-(ph — 2a^+12«e 
2a^p^-4«X2a^+2ap)+48aV >=-.0. 
+ (p . — 4:ae(cc—(3){(3 — y)(y—u) 
+ — a^P^y^ + 4«eaPy 2a — 1 6aV2aP + 6 4«V 
10. In this equation the coefficient of is 
-4 «6.8(B-AXC-B)(A-C) 
= 32«e (A-B)(B-C)(C-A); 
or, neglecting the multiplier a, it is 
-32.1.2.3.4(1-2X1--3X1-4X2-3X2-4)(3-4), 
which is the value for 5 = 0, of 
-32(1-2)(1-3X1-4X1-5X2-3X2-4X2-5)(3-4X3-5X4-5), 
i. e. the coefficient in question is 
-32.25v/5-v/«y'^+&c. = -800v'5\/«y'+&c., 
where ay^-f-i&c. denotes the discriminant of the denumerate form 
(a, b, c, d, e,fjv. If. 
20. The remaining coefficients are rational functions of a, b, c, d, e, which have to be 
completed by the introduction of the terms inf. We have 
Coeff. 
= - (2aX =-c^ 
+ 22ap _j_2(-16«^-l-4M-cy 
+ 12«^ -\-Vlae. 
CoefF. 
(2aPX = ( — 16ae-j-4M— 
— 2aPy2a —‘lc{V^ace—8ad^—8b‘^e-\-^bcd—&) 
— 4ac(2aX — 4«c^e 
+ 4«e(2aP) -\-4:a€{~l^ae-\-^bd—c^) 
+ 48aV +48aV. 
