340 Mr. J. C. Douglas's Reply to Mr. Templeton's 



tion of a from the south pole of the spheroid. We have thus 



tan (a) _ ^/haP + hz* _ ^sin 2 X + cos 2 X 

 tan ct 



\J dx* + ~ Sz* ysin 2 X + A* cos 2 \ 



V 



1 + cot 2 X _ sin V 

 1 + cot 2 X' ~ sEaT' 

 which is the required relation. 

 The foregoing equations, 



cos X f sin a = cos V , tan X ; = -r- tan X, 



A 



. , . sin X' 

 tan (a) =— — —tana, 

 w smX 



determine in the stereographic projection the inclination (a) to 

 the radius, or projection of the meridian, of the geodesic line (pa- 

 rametric latitude of vertex =/') at the point the parametric lati- 

 tude of which is =X' ; viz. they enable the construction (in the 

 projection) of the direction of the successive elements of the 

 geodesic line. There would be no difficulty in performing the 

 construction geometrically; but it would, I think, be more con- 

 venient to calculate (a) numerically for a given value of I' and 

 for the successive values of X'. Observe that for X' = we have 



i n ™^ 7f ^ xi sinX' tanX' C 



(as above) 90°— «=/'. and then . \ =- — — = — , conse- 

 v J sinX tanX A 



C A 



quently tan (a) = j cot V : but we have also cot V = p cot /, so 



that this equation becomes tan (a) = cot/, or we have 90°— (a) = /; 

 viz., in the projection, the geodesic line cuts the equator at an 

 angle /=the normal latitude of the vertex of the geodesic line. 



The preceding formulae and results have enabled me to con- 

 struct a drawing, on a large scale, of the stereographic projection 

 of the geodesic lines for the spheroid, polar axis = J equatorial 

 axis. 



XLIV. Reply to Mr. Templeton's " Remarks suggested by Mr. 

 Douglas's Account of a New Optometer" By J. C. Dou- 

 glas, Government Science Teacher, Assistant Superintendent 

 East -India Government Telegraph Department*. 



UNFORTUNATELY Mr. Templeton has misunderstood my 

 description of the optometer principle ; his proof that the 

 indications of an instrument on this principle are fallacious is 

 therefore not applicable. I did not give exact measurements, as 

 such were not necessary to a description of the principle; I 

 * Communicated by the Author. 



