Mr. A. Cayley on the Theory of Logarithms. 



859 



the other two coordinates being x and y. If x be negative and 

 y be indefinitely small and positive, then 6= +1, and we have 

 f=7r ; but if {oo being still negative) y be indefinitely small and 

 negative, then 6=— 1, and therefore f= — tt, i. e. there is a 

 break or abrupt increment 27r of the coordinate f in passing 

 across the negative part of the axis of as from a negative to a 

 positive value of y, or as we have before called it, from below to 

 above ; this is the only discontinuity in the surface, the form of 

 the surface being, in fact, what is intended to be represented in 

 the annexed figure. 



.te;[.(^^%+ 



lfii|t-"^8i: iioda 



'F"^ 





iif tamer 

 Suppose now sr=o(^ -{-yHj z=x + yij and consider the definite 

 integral 



fjcf :>if> ..M 0' f ft'^it ^^h; 





■,0 '«4-,u 



\t,3l3rfw 



the path beinsr, as before, a riffht line. We have by the equiva- 

 lent analytical definition, i,^^• • ^ , ^^^ ^^ ^^^j 



where the new variable r is real. And in like manner consider- 

 ing the integral 





« t f rf* it fM^ i n '^»*% i/7.i?»^ f 



d^"^^ yfi ^9f3Xfw 



the path being in this case also a right line, we have ^ 



z' 



C^du fs' \ C' dr 



:5 s. 



^tjtahis^ a lo ^aum^i^^-^. 





