Jan. 1 6, 1890] 



NATURE 



259 



and F"(4)) = F'((^)r^^ ;/»(sec 4))" + '+ m tan <^T + F(<^)R''"/<'« i'^''(sec <^)«+^ + -^(« + l>"(sec <^)'' + Manf + >w(sec <^)2~l, 

 or 



F"(<^) = F(</))r'^'''?«2«(sec4))2« +2 + 2-^-7<"(sec<^)« + i tan <p + ;«2(sec (^)- - nf\ 



* ":/<2»(sec 4))2« +- +^(« + i)«"(sec<j))« + 1 tan 4> + ^(sec ^)- 



= F(«^)-^ — (/ + «)«<2«(sec 0)-« + - + -(2/« + « + i)«» (sec ^)" + ' tan A + w(w + i)(sec<f)- - w^ l 

 '- iT . .4'' i 



Since 



~ = - u ^ (sec <^)" + \ 

 this last expression may be put under the form — 



F"(0) = F(4)) I /(/ + »)(^'l^' + ik^rn + u + O ( -" ) tan f + m {m + i) (sec i>)- - m-\ . 

 Hence, by the above lemma, 



rVsec<^"V(^ = (a - i8^F(7) j i + h (« - ^^'[^(^ + «/ jj' Y + ^^2«i + « + i) (-^) tan7 + m{m + i)(sec7)- -»'']} 



= {a - B) 2/, (sec 7)"'{ i + V^ (« -^".(as before) | 



where ( ' | denotes what — becomes when w = o, or when 7 is substituted for <t>, and «„ for //, that is — 



f du\ k n , ^« + 1 



I — r — - " (sec 7) ^ . 



The factor «'^^ may be eliminated from this expression, and the expression itself simplified, by means of the formula — 



- = (« - 



)/ — / .71— I ^ 



'1 P 



^Vd,=^(^^ fw^sec.^r'''^^, 



//" 



for, putting in = « + i in the above expression, we have — 



/*■' «' (sec <pT + V^) =: (a - ^);/o (sec 7)" ^ ' { i + sVla - )8)-r/(/+ "/^ Y + 3A« + i)f '^ V^n 7 + « +"1" « + 2 (sec 7)^ - (« + l) 2 jj- 



Hence 



1^ u' (sec <^)"./<?> - J; ./ (sec ^f + ^ d.p, or J^ ./ (sec ,|.)'« ^.^ - ^_:^-^(_i-, - ^_i-,) 



= (sec 7)'" — " ~ ^ I I + V? (a - j8)'^ 2/(w - « - i) [—t] tan 7 + w« - « - I w + « + 2 (sec 7)- - /« - ;/ - Iw + w+i | • 

 It will be noticed that the term involving ( ** ) has disappeared by this division. 

 Now make m = 2, and this formula becomes — 

 \\^ (sec .^f d<p = ^/^^ - ~r) (co« 7)" - ^ { I - .M« - 3){2/(« - I )(^^)^ tan 7 +^i ^4 (sec 7)= - ^i ilTs]}- 

 Divide throughout by g, and put / = 2, then, from before, 



X = ' -, f -i- - -i-'j (cos 7)« -^ I - ^^-- (« - iS)- [4 (^) tan 7 + (« + 4) (sec 7)- - ^"Tl] }. 

 >&(« - 2)\4'''-"'' /"-V I 24 ^ L \ud<ph J) 



Similarly, divide throughout by^, and put I = l, then — 



T= ' (-^ _^i-)(cos7)«-M» -^^^I(a - P'fl^i-^h;) tan7 + (;^ + 4)(sec7)--» + 3l{- 



Lastly, let 



so that 

 then 

 and 

 Hence 



F{(f>) = u' (sec (p)'" tan <p = A'P) tan <?» suppose, 

 /{<p) = M'(sec <l>)"' ; 

 F'(<^) = /'(<^) tan t + / (<?>) (sec ^)2, 

 F"(<^) =/"(^) tart <t> + 2/\<t>) (sec (pf + 2j{<p) (sec <p)- tan ^. 



['F{ip)d<p = {a- j8){F(7) + tjIt (« - ^)"^ F"(7)l approximately, 

 = (« - J3) |/(7) tan 7 + A(« - )3F[/'(7) tan 7 + 2/(7) (sec 7)2 + 2/(7) (sec 7)2 tan 7] ^ ; 

 f'^/{<t>)d<p = (a - 18) 1/(7) + ^i"' - J8)y "(7)!- approximately ; 



f'F{<l>)d,p ^ ['A<PW<P = tan 7 +" V5(« - -^^'P/T^ '^^^^ '^^'' "'' ^^'^'^ '^'^' t^" T J ' 

 in which the term involving/"(7) has disappeared. 



also 



and therefore 



