256 



Archdeacon Pratt on the effect of 



00"+ ( ^ \^ — 5 — ^— ) = a minimum ; 



.• . {(P— Q cos 2jny + (P + Q cos 2mf}U = (P + Q cos 2m)ls ; 

 P + Qcos2m Is P — Q cos 2m hs 



p2 + Q2cos'2m 2 ' p2+Q'cos2 2m 2' 



^^'+^^- F+Q^cos^2m T' 

 This is least when m—0 and 90° ; then 



aa=-M!£, ^6=Z±Q?^, la^lh=-^. 

 P + 2' P^^ + Q^ 2' F + 



Let one of the two ellipses be equal to the mean ellipse of the earth's 

 figure, a and h being the semiaxes, and la and the excess (or defect, if 

 negative) of the semiaxes of the other ellipse. The first ellipse is not ne- 

 cessarily the mean ellipse itself, but is only equal to it in dimensions, and 

 parallel to it in position ; for the actual arc may lie above or below the 

 mean ellipse. The result of this is, that the arc of the mean elHpse which 

 corresponds with s of the actual arc will not necessarily have precisely the 

 same middle latitude, although the chord c is of the same length. But as 

 the middle latitude will differ only by a quantity of tlie order of the ellip- 

 ticity, this difference will not appear in the result, because we neglect the 

 square of the ellipticity. 



I will now make the extravagant supposition that the ellipse to which 

 the arc actually belongs deviates from the form of the mean ellipse so much 

 that la'^lb=\^ miles, the whole compression of the earth's figure. On 

 this supposition I will find how large the arc may be so as not to produce 

 a difference in length greater than V\ 



Put la--lb = \2>, a5=l" = 0-0193 mile (1° being 69*5 miles), 

 (P2+Q0^Q=0-0193-M3=0-0015, 



or 



tan -Y + i tan- - (1 - cos X)-= 0-00075 tan ^1 (1 - cos X). 

 \2 2/ 4 2 2 



A slight inspection of this equation shows that X must be small. Expand 

 in powers of X ; then 



G + (^y=^*^^^^' (g =0-00135 ; 



.-. X=0-22 (in arc) =0-22 x57°-3 (in degrees) = 12°' 6. 

 This shows that in an arc of meridian as much as twelve degrees and a 

 half in length, it would require a departure from the mean ellipse equal to 

 the whole actual compression of the pole of the earth in order to produce 

 so slight a difference in the length as 1". Hence we may conclude that the 

 difference in length between the mean arc and the actual arc, joining any 

 two places on the same meridian, is an insensible quantity, since an extra- 

 vagant hypothesis regarding the departure of the form from the mean form 

 will not produce a difference in length of more than 1". This being the 



