412 Mr DAVIES on the Equations of Loci 



I cos 6 ( cos 8 cos a sin e -+- sin 8 cos e) ) 



cot(p = coseca cosec6-{ A a r> r (9- 



I + sin 6 ( sin p cos a sin _|_ cos 8 cos e) ) 



This expression must be substituted for the (9.) of that paper. In that pa- 

 per, I had taken only the + in deriving equation (7.) ; and in obtaining (8.) 

 I had employed the equation 



sin K \J 1 cos 2 K instead of sin K = V 1 cos 2 K. 



The formula thus obtained was incomplete in an analytical point of view ; 

 and the complete analytical solution was redundant in its application to the 

 confined geometrical aspect under which I then viewed it. Unfortunately, 

 too, the particular algebraical case, and the particular geometrical one which 

 I separately considered, were not those which corresponded to one another ; 

 and consequently, by attaching them together, the result which thus pre- 

 sented itself was not merely incomplete, but it was erroneous. No conse- 

 quences were deduced from it, however, in that paper ; and hence no incon- 

 venience has arisen, except to myself, in being obliged to recalculate those 

 expressions into which this equation entered. The rule for the signs, then, 

 is this : 



When the angle e is estimated from the branch of the meridian LP 

 between the given point and the pole of reference, the sign of cos e is + ; 

 and when from the opposite branch LP', then is to be employed. 



Thiis, if the angle e be P'LD, the sign + or is to be used accord- 

 ing as the pole of reference is P' or P. 



