208 STRENGTH OF MATERIALS 



The problem now consists in finding the value of the variable angle 

 x for which f is a maximum, which may be expressed symbolically 

 by the conditions 



= and - < 0. 

 dx dx* 



In order to reduce the expression for P 1 to a form more suitable for 

 differentiation, we make use of the following identity. 



cos (a x) cos (a a>) 

 cot (a x) cot /a a>) = 



sm(a-x) sin(a-o>) 



cos (a x) sin (a a?) cos (a a>) sin (a x) 



sin (a x) sin (a a>) 

 sin (JG co) 



sin (a x) sin (a a>) ' 

 whence 



sin (x w) = sin (a x) sin (a a)) [cot (a x) cot (a a>)]. 



Similarly, 



sin(# 13)= sin (a it') sin (a )[cot(a x) cot (a /3)], 

 and 



sin(# + ft> + f #)= sin(a #)sin(o> + f)[cot(a x) cot(o> -f ?)] 

 Substituting these values in the expression for P 1 , the latter becomes 

 , _ wh*sin(a co) cot( x) cot(a <o>) 



2sin 2 o:sin(cr)4-f) [cot(a x) cot(a y8)][c.t <<< , ) + c<>t (o>+f)] 



Now the terms in this expression which contain tin- variable x are 

 all of the same form, namely, cot (a x). This term may therefore 

 be replaced by a new variable y, and the remaining terms by letters 



denoting constants. Thus let 



wh* sin (a o>) 

 cot(a-x)=y, -! = I. 







sura sin(a) + f) 

 cot (a co)= B, cot (a j3)= C, cot (o> -f J) = Z>. 



Then 



Equating to zero the first derivative of P' with respect to y, we hav- 



dpf - A (y- )(y + -P)-(y - *)(y - ^-(y - ^)(y + ^) = . 



dy (y - C)\y + D) 2 



whence the condition for a maximum is 



D). 



