Mr. HOPKINS, ON THE MOTION OF GLACIERS. 71 



2 re Zirt' 2 /•« ^ , 2Trt' 



A cos ( = ~ ipv) cos — -— at , J sin e = - d)(t )s\n ——- dt , 



6 Jo ^ 0Jor\' Q 



2 re i-n-t' , , ^ . 2 /-a , , . inf' 



^,cose, = -j^ <p (t } cos -—dt, ^.sine, = -j^ (f ) sin -— rf^ 



&c. = &c. &c. = &c. 



~ r« 1 /vv 2(w+I)7r/'' , ^ . 2 re , ^ „ . a(n + l)7r#' , , 

 j4„cose„=- / <p(t)cos dt , A„sin€,, = - <p(t)s\n —^ — dt , 



kc. = &c. &c. = &c. 



In the application of these formulse to the case before us we have 



B= f^(l>{t')dt'+ f^y(t')dt' 



= f]^+ Ccos Uirt' + '^]\df' 



V c 



i ~ TT ' 



^cos e = 2 f'\^+ Ccos Uwt' + -jicos.27r/' dt' 



= 2 r Iv cos ^Trt' +- cos I iTrt' + ~]>dt' 

 = 

 A sin e = 2 /""I r+ Ccos l2Trt' + -j > sin27r#'d^' 



= 2 ^' rFsiii27r#' +- I sin Utrt' + j]- sin -il rf?' 



_ 2F C 



Hence, e = -, ^ = . 



2 TT 2 



.Vlso tal<ing the general term, 



^„C0S6„ = 2j'-\v + C COS faTTi'' + ^jl cos 2(n + ])Trf'df' 



I M I'cos 2 (n + 1) Trt' + — fcos 1 2 (n + 2) irt' + -) + cos Ismrt' - ^) [ d^' 



2 



' when n is even ,• 

 C 1 



when n is odd. 



IT w (« + 2) 



^„ sin €„ = 2 /" |r+ Ccos (2 7r<' + -) i sin 2 (w + ])7rt'dt' 



= 2 ^ JFsin 2(m + l)fl-/'+ - fsin (2 (n + 2) n f' + -j + sin |2// tt/' - ^)1 > rf<' 



