traced upon the Surface of the Sphere. 311 



concave hemisphere P'Q'L/, as well as the area of the branch LM^O de- 

 scribed. And finally, during the remaining quadrant of increase of (f>, the 

 remaining half P'E'L of the concave hemisphere, as well as the portion 

 LOPQL will be passed over by the radius vector. Adding these together, 

 we have 



PM,O + POM,,LQP + P'Q'L' + LM/) + P'L'E' + LOPQL, or 

 (PM,0 + LOPQL) + (FOM,,LQP + LM,p) + P'Q'L' + P'L'E'. 



These make up four spherical quadrantal lines, or the whole surface of the 

 sphere : and such would have been the anticipated result of taking < be- 

 tween limits and 2 T. But this is not the case : for then we have only 



or to a single hemisphere. How is this to be explained ? I have formed to 

 myself a theory of the apparent anomaly, but it is connected with so many 

 considerations, that the discussion would take up more room than I can allow 

 myself in the present paper*. Whilst, therefore, I defer this discussion till 

 a future time, I think it necessary to state, that the difficulty may always 

 be evaded by transposing the origin of co-ordinates to the nodal point of the 

 curve, as, for instance, to the point O in the preceding figures. The for- 

 mulas of transformation adapted to this case are given in (XIV. 3), and the 

 process is too simple to need farther illustration. 



If, however, the figure be composed (as in the present case) of four equal 

 branches, it will be sufficient to find the area of one, and take its quadruple 

 for the whole area. This is analogous to those processes in plane curves, 

 where we find the areas of the branches above and below the axis of x sepa- 

 rately, and take (in symmetrical curves) double of one of them for the whole 

 area. 



* Some preliminary considerations intimately connected with this subject, are given 

 in Note (B) at the end. 



