Mr R. B. HAYWARD, ON A DIRECT METHOD OF ESTIMATING 



Let U, If, U" and F, F', F" be three consecutive positions of U and F respectively, and 

 K, K' those poles of FF", F'F" respectively, (considered as arcs of great circles) about which 

 positive rotation brings F to J", and F' to F". We know that IT lies on the arc UF between 

 U and F, and V" on the arc ITF' between If and 

 F'. Also it is plain that F' is the pole of KK', and 

 therefore that KK' measures the angle ofcontingence 

 between the consecutive elements FF', F'F": in 

 fact, the loci of K and F are so connected that the 

 elementary arcs of the one are equal to the angles 

 of contingence of the other, and vice versa. 



Suppose the locus of F to be defined by ele- 

 ments, corresponding to what Dr Whewell has 

 named in plane curves intrinsic elements, that is, 



by elements a, e such that the elementary arc FF' = da, and the angle of contingence between 

 FF' and F'F'' = de : and suppose the locus of U defined in like manner, so that UU' = d(p, 

 and the angle of contingence FU' V" = dri. Also let UF = in, and angle UFF' = v. 



Now let OP coincide always with OF. Then will Up=u cos m. and / being taken to 



coincide with k, Q = — , and therefore equation (3) becomes 



dt 



d -. . da . 



J- (u cos fx) =f-u-—. sm n . sin KU.sin KUF. 



dt dt 



But sin kU . sin kUF = sin kFU = sin ( v] = 



cos V, 



and— = /cos (IX ; 



/ . 

 ■cosi/ + -smu= 0. 

 u 



.(1). 



therefore we obtain after reduction 



dfi da 

 dt ~ dt 

 Again let OP coincide always with OK, then 



Up'=u cos UK = M sin M • cos UFK = m sin ju , sin v, 

 and I may be taken to coincide with F' or ultimately with F, so that (3) becomes, 

 de\ 



(q being = ^), 



^ . . de 



— (m sm jui .sin v) = -u-~.sm FU . sin UK. sin FUK 



dt 



or after reduction 



dt 



de . 

 = — M — - Sin fj. . cos V, 

 dt 



d/u. 



dfi tdv de\ 

 dt"^ [dt'^dtj' 



tan a f . 



h ■- Sin ^1 = 0. 



tan V u 



.(2). 



The equations (l) and (2) together with the two equations (B) serve to determine (after 

 du d(p , , da de 



dt "" 



eliminating fx and v) -£ and"-^, when f, -jj, ~ are given, that is, when the intensity and 



dt 



dt dt 



