35 



sin mx — ax + Ba;^ + cx'^ + &c. 

 justifying this assumption on the ground that x and sin 7nx 

 vanish together ; which can be considered valid only so long 

 as w — 00 is excluded. In fact, whether we seek the deve- 

 lopment of sin mx after the manner of Lagrange, or by the 

 theorem of Maclaurin, it is essential to the very nature of the 

 investigation that the unknown coeflScients a, b, c, &c. be all 

 assumed to be finite. We cannot conclude, therefore, from 



^ , . , dsmmx , 



Lagrange s reasoning, that — zz m cos mx, when m is 



infinite : and similar considerations forbid the conclusion that 



; = — m sin mx, in like circumstances. The method of 



ax . 



limits equally militates against such a conclusion ; thus, if the 

 function were sin x, we should have 



sin (« + ^) — sin a; = 2 sin \ h cos (a; + i h), 

 or 



sin {x •\-K) — sin x sin \h 



hh 



cos (a; 4- 2 ^) ; 



and since ■ ^ ^ zz 1, in the limit, or when A = 0, we should 



usin X 

 safely infer that — -^ — = cos x. But, by proceeding in like 



manner with sin?wa;, we should have 



%m (mx -\- mh) — sin ma; ?>vahmh , . , ,x 



— ^ z-^ = m — = — ;— cos (mx + ^mh), 



h ^mh • - / 



from which, if m be infinite, it could not be inferred that 

 — -j =z m cos mx ; since we have no right to affirm that 



1^1 tends to 1, as A diminishes, and finally terminates in 



that value when A = ; nor that, in like circumstances, 



cos {mx + i mK) ■=. cos mx. We have nothing to justify the 



, siniA , %mhmh , , ,. . 



assertion that , , and -^j — j- are the same at the limits 

 ^ h ^mn 



