302 



SUPPLEMENT TO CHAP. XIV. 



[CHAP. IV. 



In any case in which the approximate formulae for c\ and e\ are used, it 

 i* well to check the results obtained, by plotting the intersection and direc- 

 tion curves, and drawing for each position of the weight the reactions, and 

 thus determining Ci and C 3 , as shown in Fig. 92, PI. 24. 



20 (&;. Traiislorniutiou Series. We have in the preceding repeat- 

 edly made use of series in the transformation of angular functions, such as 

 sin, cos, etc., into functions of the arc itself. We group here, for conveni- 

 ence of reference, the series thus used : 



sn x = x ( - 



cos = 1 i a? + & x* - 



cos 2 * = 1 - 

 sin a; cos* = x(l j- a? + -ft- ic 4 



a 8 as^aoo * 10 + 



* 10 



cos 3 * = 1 - ?** + I* 4 - AV* 6 + liU^se 8 - Tflftr* 10 + 

 tan * = * (1 + i 



1 * 



cot* = -- -7J- 

 * o 



sinzsiny = xy [1 - $ (z 2 + /) + ^ (S* 4 + 10* 2 y 1 + 3 y 4 ) - TTuVir ( + 

 + 7**y* + y 8 ) + -rrri^To {5 a? + 60 *V + 126 *V + 60 a;' 2 y 6 + 5 y 8 )...] 



+ y 6 ) + 

 sin * cos y=x [1- 



28**^ + 70 *V + 

 Tr (a) 4 -1-10 * 2 y J +5 



8 + y*) -.. + 



