Prof. Cay ley on Differential Equations and Umbilici. 443 



and hence, writing 



y = (b + cv){v*-l)+2{f+gv)v=c{v-u)(v-f3){v-r / ), 



and 



c(v*-l) + 2gv _ c{v*-l) + 2gv A B C 



V ~~ c(v— a)(y— fi)(v— y) ~ v — a. v—ft v—<y 9 



so that 



A _ c(a 2 - l)+2ff« 



c(u*-l)+2g* + 2{f+{b+g)* + c a ?\ 



with the like values for B and C— values which are such that 

 A + B + C = l, 



the integral equation is 



const. =#(?;— w)(w— a) -A (v— /3)~ B (v— y)~ c , 



, ... .. " - -. i c(v— a)(v— /3)(v— 7) 

 or, substituting for v— w its value = — — . g__-,x . A —3 



const. =^{c(i; 2 -l)+2^}" 1 (v-a) 1 ">-/3) 1 - B (^-7) 1 - c . 

 But 



5 + cm 

 if for shortness TJ=(& + cw) 2 + (/+#w) 2 , and thence 



" ,,«_. 2(/+^) a + (ft + cm) 2 +2(/+^)y / U 

 (6 + cm) 2 

 and 



(6 + cw) 2 _ 



B _ g= -(/+gn)--/U-«(6 + «i) &c _ 



6 + CM 



Substituting these values, and observing that the exponent of 

 b + cuis (-2 + l-A + l-B + l-C = l-A-B-C) = 0, the 

 integral equation is 



const. = %(f-\~gu-\-\/XJ)~ 1 



x(f+gu + *(b + cu)+\/vy-Xf+gu + !3{b + cu)+x/vy- B 



(/+^ + 7 (ft + ^)+v / tJ) 1 - C . 

 Or, observing that the exponent I of a? is 



= _1 + (1_A) + (1-B) + (1-C), 



and putting for shortness □ = (fis+gy) i + (bx + cy) 2 i the inte- 



2G2 * 



