15 



sin 3 <f> ( 1 sin 4 <£ 1 d <f> 



j o Sa 2 \ 16 a 4 / 



= (expanding by the Binomial Theorem) 



/ : 



8 a 2 



* / I 



( 



74 



sin 3 <f> 1 + sin 4 <£ + J d <f> 



32 a 4 / 



Now integrate each term separately, and we get 

 I s I 7 



7/ — 2 i 6 • 4 . 2. i 



y — s + T53 + 



8 a 2 8 x32a 6 



P W 1 V W* 



= + . + 



48 E I 560 64 E 3 P 

 The first term of this series is the deflection as usually 

 computed, so that if we denote this by d, we have 



108 d s 



y = d + .— 



35 I 2 

 The ratio of the error made by the ordinary computation 

 to d is thus : 



108 d 2 



35 I 2 

 d 

 Since — is usually in practical calculations something 

 I 



considerably less than "01, it follows that the error is not 

 worth taking account of and is really quite negligible. 



Similar results apply in other cases of the deflections of 

 beams, so that it seems that in such computations the ordinary 

 method of calculation is quite good enough. It appears that 

 the usual mode of solution can only lead to errors of a serious 



nature, when, in addition to ( -f- ) the equation to be solved 



V ax J 



contains other terms that are themselves of a small order 



dy 

 comparable to — , and in problems on the deflections of beams 



dx 



this is not the case. 



