Jan, 1 6, 1890] NATURE 261 



The above expressions for X and Y may be transformed by" introducing this angle 7 into them instead of 7, thus — 

 (cos 7)''-^ = (cos 7)''-i - (« - I) (cos 7)«-2 sin 7 \}j^ (« - /8) + i(« - &f tan 7J 

 = (cos7)«-^{i-^'^-^(«-j8)tan7- '^-"J- (« _ /3)Mtan 7)" } • 



^=>^)(^-/^0^-^^)"-^ { ' -~^V-m«^2(sec7)^-«-::i]}. 



Y = X tan 7 = ^— ^ (-^_-7, - •^-,j (cos 7)" -'^ sin 7 { i - ^^ (a - )8)2 [^^ (sec 7)2 - ^^] } J 



^ = i(^(r^2-/^)(^°^"^^"-^'Q' 



Y = — ( - — — - ) (cos 7)" — - sin 7 Q ; 



Hence we find — 

 and 



n - \ 



Q being = i - — _ (« - ^)2 [« - 2 (sec 7)2 - « - 3']. 

 24 

 Similarly, if 



7'= 7 + ^^J:_£^(a _ ;8) + 1 (« _ ^)2 tan 7, 



we have 



and therefore 



T = 



(cos 7')"-"^ = (cos7)«-i - (n - I) (cos7)»-2siiiX 1/ £. („ - )3) + \{a. - )3)2 tan 7 ~| ; 



L"/ + ? J 



= (cos7)»-i 1 1 - ^±/^X (« _ ^) tan 7 - 'L^l-i,o. - ^f (tan 7)' } ; 

 \ O p ->r q 4 -' 



where Q has the same value as before. 



Hence the values of X, Y, and T are as stated in my postscript to Mr. Niven's paper. 



Although the method of finding the expressions for X and T given above, is perhaps the plainest and most straightforward^that 

 can be taken, the following leads to simpler operations. 



Let y(<^) = M^(sec <^)«+i. 



Then if^d^ = /'^'(sec ^)"+i^^ = | /"«-;- «-i ^^ d(p by equation (i) 



= —-—§- — «'-« + const. 

 k(l — n) 

 Hence 



Now let 



F(<^) =f[<p){sQc <p)"' = «'(sec <^)'«+«+i, 

 then 



F'(^) =/'{f){sec (p)'" + mf{<p){%tc </>) '«tan <p, 

 and 



F"(f) =/"(<^)(sec ^)'« + 2OT/'((^)(sec ^)"'tan <|) + OT/((/))[;«(sec <^)"'(tan </>)2 + (sec <^)'«+2] 



= /"(4>)(sec <^)»* + 2m/'(<p)(sec <^)»»tan <^ +m/[^)[m + i(sec^)"'+2 -»»(sec<^)*»]. 

 Hence, by the lemma, 



rF(<^)^<^ - (a-i8){F(7) + ^V(« " 0)'F"{y)} 



= (a - ^)|/(7)(sec7)'« + ^(o - i8)-(sec 7)"'[/"(7) + 2w/'(7)tan y + m/{y)[m + i(sec 7)2 - m]\]j 



= (a - ;8)(sec 7)"' ^/{y) + ^\{a - m/"(y)+ 2mf {y) tan 7 + ;;;y(7)[^TT(sec 7)^ - w]] j • 

 But from above 



-J—lf-^n _ al-^n) ^ [ /{<t>)d<p, 



k{l - ny ^ J fi 



= (« - i8){/(7) + ^V(« - ^)y"(7)}. 

 Hence, by division, 



j F{<p)df-^^-j^{f »-/-:«) = (sec 7)'« j i + ^a -- py\_ zm-^—tuny + fu{m + i(sec 7)2 - w) J | . 



