310 REPORT — 1873. 



a being a whole number. 



We see at once that (/3 and y being any whole numbers) 



03, s{^ + (^ logj^ + 2yA, Mr + 2/3A+V log,?} = e-^03, 3(^> '^)' 

 where wc have for M 



Now, then, consider the quantity 



tj'^log q-\-'w'^ log p — iAviv 

 € log^i^log^!?-4A^ ^^^^ ^(v, w) ; 



aud substitute in this formula v + /3 log^p + 2yA for v, and w + y log^ g' + 2/5A 

 for IV, and the formula becomes 



iPlog q-\-w-lQg p—4:Avw 



! 1 UM 



e log^i^log^2-4A^ 03^3(,; + /31og,i> + 2yA,t. + ylog^2 + 2/3A), 



v^log ^+w^log p—4Avw 

 or e log^i>log^!?-4A^ 0^^ ^(v, w), 



and therefore remains imchanged. We shall soon meet with a series of 

 functions similar to (p^ J^v, w) and doubly penodic ; this theorem will enable 

 us to show that the ratios of these functions are also doubly periodic with 

 different periods (p. 411). 



^^^ vlog^q-2Atv ^^ wlog^p-2Av 



log,i> log, ? — 4A' ■ log j:. log, q - 4A''' 



then 



^2 logjjlog 5r-4A= log «log o-4A^ 



:+ w-^ , , ■ ^ iJ. (W+»^)= 



eK^ log,i> ^^^^(„^^)=2,e 1%^ 



-00 

 (v+2nAy^ 



.e ^%^ 03(t; + 2A»,p), (1) 



ufl log,i'log^!?-4A^_ log^?>log^-4A= 



+ — w^^ — ^ . . ^ w^^ — (^+«0' 



-co 



{■w+2mAy 

 log 9 



e '"».? 03(m; + 2Ah7, ?), (2) 



* To prove (his, write down the fully expanded form of 



f'V„ 3(''+/3 log^ i) + 2yA, «'+2/3A+y log^?). 



