VOL. LXXXVI.] PHILOSOPHICAL TRANSACTIONS. 711 



This may also be deduced from the expression z=t — -j^t^ + ^t^ — -^t^-^ &c. 

 ■^ = I — ("^ M t-^ — t'^ M &c. = = — — and therefore ~ = sec.^ t. 



dt ' ' 1 + « sec* t dz 



Hence also we deduce the differential ratios of all successive orders. 



(/ . sin. z d , COS. s 



VIZ. — ; — = cos. z — ■, — = — Sin z 



dz ~ " dz 



d' sin. z ■ d^ cos. z 



—J,- = — Sin. z -~dl~ — ~ '^^^•^ 



<P sin. z d' COS. z , . 



rfz' ' ^ dz' ~r • ^ 



d^ sin. ; , . d* cos. z , 



— ;- — = + Sin. z — J 4— = + cos.z 



dz* ' dz* ' 



tf' sin. z , d^ COS. 2 



— -.-- — = + COS. z ,j— = — Sin. z 



dz^ — I — • - rf,s 



In the 3d part is treated the analogy between logarithms and the trigonome- 

 tric functions of circular arcs. 



§ 17. By M, — ^- = ^ + r- =^^ + ri— "■" + TTTTT^ ^' + ^'=- 



And (§ 13) cos. ^ = 1 - J.- ,^ + — '- ^^ - — J_ ," 4- &c. 



These expressions differ only by the signs of the alternate terms, which 



contain the oddly even powers of z: therefore, if in the former we change the 



sign of z^, by substituting — z'^ for z', or z-v/ — 1 for z, we shall obtain the 



2d, whence we have been said to represent the cos. z under the imaginary ex- 



ponential form, cos. z = 



Likewise (§ 4), ^— = z + — 2~~3 ""^ + T~27."5 ^^ + TTiT.Tr ^' ^^- 



And sin. z = z - —^3 z3 + — J.— z^_— i--^:,7+_^_^9_&c. 



If in the 2d member of the former equation we substitute %V ~\ for z, and 



divide the result by \/ — 1, we obtain the 2d member of the 2d equation. Hence 



we have been said to represent the sin. z under the imaginary exponential form. 



e 



Sin. z = 



2 V-l 



1 



Hence, tang, z = -r_- = ^^T^i -«,/-i ; and < v^ - 1 = ^^73^ n^T^.- 



Therefore e«^-' : e- « V-' = 1 -j- < ^z — 1 : 1 — < / — I, 



ore^'^v^'': 1 =l+^v^— 1-1 — ^V^— 1; and 



This formula might also be deduced from the 2 expressions, 

 .. log. l±l==v + ^v^ + \v' ^- \v' +^v' + &c. 

 and z = i - 4- «' + X «' - i i' + i '' - &c. 



