362 



Prof. G. M. Minchin on the Magnetic 



the lines of force by the mathematics of orthogonal trajec- 

 tories. 



The magnitude of the conical angle subtended at any point 

 by a given circle can be expressed in finite terms by means 

 of complete elliptic integrals of the third kind. The para- 

 meter involved in these integrals will depend on the way in 

 which they are taken. 



If a sphere of unit radius is described round P as centre, 

 and lines are drawn from P to the points on the circum- 

 ference of the given circle, BMAI, fig. 4, these lines will 

 intercept on the sphere a spherical ellipse, bmai, whose area 

 is the conical angle subtended by the circle at P. The minor 

 axis of this ellipse is the great circular arc ab determined by 

 the lines PA, PB, while the major axis, mi, is determined by 

 the chord, MI, of the circle which subtends a maximum angle, 

 MPI, at P. This line is determined by drawing the bisector, 



Fig. 4. 



PC, of the angle BPA, meeting BA in C; then MI is the 

 chord through perpendicular to the plane BAP. The 

 point c in which PC meets the surface of the sphere is the 

 centre of the spherical ellipse. 



Now, given any curve, mpi, fig. 5, on a sphere of unit 



Fig. 5. 





radius, its area is J(l— cos 6)d<f>, where, if o is any point on 

 the sphere inside the area, is the circular measure of the 



