in the Calculus of Functions, noticed by Mr. Babbage. 341 



but for our present purpose it is enough that equation (10.) be 

 admitted when 6 is real. 



It follows from that equation that 



- -1 y / X 



/ — . — cos . is: _ 1 y \ 



,^ V-i ^,. 0+ V^ + - = cos(^^ cos^^ — ^^j 



+ A/^sin(4-.cos~^ ,f , ) (13.) 



for an arc, whether positive or negative, has the same cosine, 

 and sin(^,cos;^-^^^^,)=^;^^ 



for if an arc be respectively in the first positive or in the first 

 negative semicircle, the sign of the sine of that arc will be 

 positive or negative accordingly. 



At this step it is important to remark the necessity that ex- 

 ists for the introduction of the expression —7==, in order to 



V Z" 



z 



y 



secure the equality of e^'^ -^ <^ ^^ 0+ »^ y^ j^ z^ to 



7* _l_ .^ J 2 



^ , =^ in all cases, for if that expression were omitted, 



V>'^ + z^ 



and therefore sin (cos~^ , ^ \ substituted for 



sin 1 /-5- cos~ /--, ^1, we should not have 



. / -1 y \ ^ 



sm I cos /- a —o I = / o "^i 

 \ 0+ v/y2 ^ ^V '^ y- + z^ 



unless 2 happened to be positive ; for since the sum of the 

 squares of the sine and cosine of any arc is equal to 1, and 

 since the sign of the sine of an arc in the first semicircle is al- 

 ways positive, the sine of cos~^ — 7 will be 

 •^ * 0+ '/y^ ^ ^2 



V.-(;^)'= 



?f + 2V ^? + 2^ 



[To be continued,] 



