916 
PROFESSOR A. CAYLEY OR THE SIHGLE 
P = 0 ^( 2 ?/) = q l (cos \rTU-\-i sin ^irii)-\-q'(cos \i tu — i sin ^ttu) 
+ 5^ (cos f 7 rw +i sin -§ 7 r«)-j- . . , 
' 2 _ 
Q = ©X 2 ^) = rp(cos \ttu — i sin ij7r») + r/(cos |7rt(+i sin #7™) 
~|-p^(cos-§7rM — 7sin-§7rtt) + . . , 
x l + i[ 
V =®^(2u)=— = jg°(cos ^7ru-j-i sin ^nu) — 5 s (cos §77 u—i sin f-n-it) 
— r/^"(cos|-7rw+ i sin-§7m) + + . • , 
1 —i 
Q / = @^(2'it) = —^=-| q'( cos i-7rt —i sin \ttu) — ^'(cos -§7 tu-\-i sin |t tu) 
■q 6 (cos \ttu — i sin f 77-w) + +••; 
and therefore also 
p=q=g*+</-+< r -, 
p,=^7f jV-r- r + f+r - 
1 
q,=^f i Do -1 i v,= l( L- 
The square set, u — 0; and x-formulce. 
37. We use the square-set, in the first instance by writing therein ?d=0 ; the 
equations become 
= aX, =cu'^t(u— x'j, 
BYi = /3X + aY, =ar$$(b—x), 
C~r = «X-/3Y, =a>m(c-x), 
l)' 2 u—j3X .— aY, = 0 ) 2 W(d—x), 
viz., the equations without their last members show that there exist functions or and 
xco 2 , linear functions of X and Y, such that ^C, J3, (Y, 13, 136, Qtc, I3d, being 
constants, the squared functions may be assumed equal to — W orx, &c., that is, 
or^(a — x), &c., respectively : the squared functions are then proportional to the values 
<3L(a—x), &c. 
To show the meaning of the factor a> 2 , observe that from any two of the equations, 
for instance the first and second, we have an equation without 10, AYpB'M=H(« — x) 
-pH3(6 — x) ; and using this to determine x, and then substituting in or = A%-r- 
(a—x), we find 
„ BAY—ABY 
