532 Notes to the Meditation on Poncelct's Theorem, 

 above given, is, as we have seen, a consequence of the equality 



log(i- fVi- g) m 2 c * d0 p og cos e '' log cos Q (cos e ^ t 2 



vl — t 2 'v* 



2 



+ log cos 6 (cos 0) 4 * 2 + &c. } . 



Hence we have 



,0 log cos 6 - 2 1og(l+\/r=^ r ) * 

 ! ff l-6 2 (cos0 ^"tt ^rT6« 



£ 



The extreme facility and brevity with which the method in the 

 text gives the value of P(c) fori indefinitely small is worthy of 

 notice, as in the usual text-books it is obtained by a very indirect 

 and circuitous process. We may obtain in like manner the value of 



i 



"d6 



(l-<?sm<9) v/l_c 2 (sin<9) 2 



on the same supposition as to c, whether 1 — e vanishes with 

 (1 — c) or remains finite when c = l. On the latter supposition, 

 the definite integral in question has for its value 



1,21, 2 



i3i lo ^+rz^iog (TT ^. 



When es=l, this becomes infinite j when e= —1, the second term 

 becomes 7 + l°g%> ana ^ e ent i r e integral is - log^ + - ; 



4 

 when e = 0, it is log j-. Subtracting the half of the latter integral 



from the former, we shall obtain 



1 



£ 



'i tf (l-8in0)» _l 

 l ™ (cos0) 3 "2' 



which is easily verified. 



By taking successively e=\/ — n, e=—\/—n, and adding 

 together the halves of the two integrals corresponding to these 

 suppositions, we obtain the ultimate value of the complete elliptic 



* From this it will readily be seen that when n is any integer we may 



obtain 1 * — lo g cos , processes of differentiation in a form in- 



J (I-* 2 (cos Wff 

 volving only algebraical and logarithmic quantities, and so, from what pre- 

 cedes, when n is any half-integer, in terms of such quantities and of com- 

 plete elliptic functions. 



