694 Prof. Horace Lamb on the 



Hence A= —1, and 



*=-■*■, • (12) 



regard beincr had to our convention as to the zero of reckoning 

 of <$>. As already stated, this is the analytical form of 

 Listing's law. 

 If we put 



*=logtani0, (13) 



the equation (7) takes the form 



The function <fi + ijr is single-valued and periodic with 

 respect to ty. The general solution, subject to the condition 

 of finiteness for 0->O is therefore 



cf) -{-^ = ^(A S cos s\jr -\-B s sin s-ifr)t~ s , . . (15) 



where 5 = 0, 1, 2, 3, The boundary condition (8) requires 



A s = 0, B s =0, (16) 



for .<?>0, whilst A = in virtue of the convention as to the 

 origin of <f>. We are thus restricted to the solution (12). 

 The formula (3) for the rotation about OP now reduces to 



- tan \6 sinews (17) 



The mean square of this over the region considered is 



ds f i 2jd ■ am / log sec ^ a 1\ 



- . • 9, 1 tan-iflsmflrffl = " 2 , t> 



4snr^aJ - \ snri« 2.) 



ds\ (18) 



For a. = 40° this = '0316 rfs 2 . The error of mean square 

 is therefore '17 Sds. The maximum error, corresponding to 

 ■0 = a, e=^7r, is tsm^otdsj or '364<:/.9 if « = 40°. 



For comparison I have calculated what would be the error 

 of mean square on the hypothesis that the eye works like an 

 altazimuth instrument. If we measure 6 from the vertical 

 OZ, and denote the azimuth by ^r, we have (/> = 0, everywhere. 

 For a displacement ds of the visual axis, in a direction making 

 an an^le e with the vertical circle, the rotation about that 

 axis is 



cot Ods sine. ■ (19) 



The mean square of the errors is therefore Ids 2 , where 



^radJ^r^ • • • <*» 



