796 
ME, A. R. FORSYTH OH THE THETA-FUHCTIONS, 
in the left-hand side; the same increase or decrease occurs in the right-hand side; 
thus if each of the X’s be increased by unity, by unity also will each of the A’s be 
increased. Hence, as immediate deductions from (23), are obtained the following 
formulae, including as particular cases, many of Posenhain’s formulae contained in the 
table at the end of his memoir. 
ikP p )+n<s>( x+1 ’ p )+n<t>h ' >+ 1 )+n $( W1 ' />+1 
v ) V* >vj \n,v J , v 
where on the right-hand side the variables are X l5 X 2 , X 3 , X 4 , Y 2 , Y 3 , Y 4 , 
and „ left „ „ „ x v a? 2 , a? 3 , a? 4 , y Y , y 2> y 3 , y 4 . 
noP’ p )+n<t>( x+1 ’ p )-nt>( x ’ p+1 )-n<s>( x+1 ’ p+1 ) 
\fl, V) \fl , v) \fl, V } \p ,V J 
=(- i ) p [ n <: y +i )+ n < +1 : * S)- n < +1 ; V+i 
(24) 
(25) 
imA p )-uJ x+1 ’ p )+n<i>h p+1 )-nJ x+1 ’ P+1 ) 
V] \fl , v) \fl, V ) \ll ,v ) 
3-Ci V l )l « 
wt>( x ’ p )-nJ x+1 ’ p )-im>( x ’ p+1 )+u<t( x+1 ’ P+1 
\/i, vj \ p, v) \/X, V ) \ p,v 
= (_i)A'+p'j~n 3 / A ’ p nof A+1 ’ p 
1 1} |_ \<r + h<r'+l) + 1 
_W A ' P+ 1 )+W A+1,P+1 
\<7+i, ,'+ir u+i, o-'+i 
■ ■ (27). 
To these must be added a Set of four obtained by increasing each of the os’s by 
K A 
— log p and each of the y’ s by log r, conjugate quarter periods: by these sub¬ 
stitutions (24) gives 
no 
A,/, )+n<5( x p’ / ’')+n<i> 
/*+!, vP \p + l,vj \p+l,v 
x ’' ,+1 \ +n3 ,p+i>p+ :L j 
\p + l, V 
