102 



The converse of this theorem may also be readily proved; viz. 



00 m 00 00 



SSa = SSa 



m n m,n m n m+n— 1, n * 



SSa = Sa + Sa + Sa +Sa + &c. 



m T\ m, n n \.n n -2, n n3, n n A.n 



= a +(a +« ) + (« +« + a ) + {a +a +a + a )+&c. 



1, 1 2, 1 2. 2 3. 1 3. 2 3, 3 4, 1 4, 2 4, 3 4, 4 



-a + a + a +a +&c. 



1, I 2, 1 3, I 4. 1 



+ a +rt + a + a + &c. i 



2, 2 3. 2 4, 2 5, 2 



+ a + a + a + a + &c. 



3, 3 4. 3 5, 3 6, 3 



+ a +« +a +a + &c. 



4, 4 9, 4 6, 4 7. 4 

 + &C. 



CD 00 00 <» 



= Sa + Sa -If Sa + Sa + &c. 



m m, \ m.m^.]. 2 m m^%, 3 m m-f 3. 4 



00 CO 

 It m m^-n— I, n 



O) CO 



= SSa 



m n m-}-n — 1, n 



The following theorem is analogous to that in the text. If r is not less 

 than s, then 



T a t m r— ^ a t — 1 8—^n 



SSa = SSa +S .Sa + S S a 



nt n m,n m n m — n-(-l, n m n t^m*-n^l, n m n r— 4i^1, n^m 



n n 



Art. 21. It is perhaps better to write {a{ instead of {a . Among 



the numerous applications of this notation^ that to continued fractions 

 may be pointed out. Thus 



