332 REPORT— 1873. 



a result substantially the same as Konlgsberger's, although it seems to me 

 that there is a misprint in his paper. 



To illustrate the Table at the bottom of page 22, I observe as foUows :— 

 Eeferring to the Table at the foot of page 20, we have 



e(V2)o.3 = eO^ + i'<+4< f, + >^ + K in? + in?, ^nl + inl) 

 = 0K-i + O, f,-i-|, |0 + iO, iO + i) 



which agrees with the expression given in the Table by Konigsberger. The 

 reader is requested to notice that Konigsberger writes ^{v^v^^ = d{v^v ), a 

 notation which we shall have occasion to recall hereafter. For illustration 

 of Table, p. 23, see remarks at the end of next section. 

 Section 3. — This section opens with an expression for 



where, it will be seen, a change of modulus is introduced. We proceed to 

 prove the theorem, as it is enunciated without demonstration. 



EecaUing the value of (0) given in Section 1, this expression is seen to 

 be equivalent to SSe^'^^''^^^ where 



+ &C. 



Now put 



which we may evidently do, provided that we sum with regard to » , u «„ 

 from to 2>. r i r-2 rf> 



Now we easily see that 



the three brackets corresponding to the three factors in the following expres- 

 sion constituting the second member of the equation 



where P = 



+ ^2(;j + l)«, + ^^"?f 2(^j5r,,,-t- . . +ppr.,,)+p(p+l)(n.^r^,,+ . .))+.... }ni, 



