f ON SEISMOLOGICAL INVESTIGATIONS. 37 



circle, but the projected point n will not be the centre. Let P repre- 

 sent the North Pole (Fig. 1), and be the centre of the projection. Then 

 if the arc PN on the sphere be A. , tihe distance Pn on the fiat will be 

 tan \ /2. Let S and E be the points where the circle with radius A 

 cuts the meridian PS, so thatPS=X-A, and PE=X+A: then the 

 corresponding points s and r are given by 



Ps = tan PS/2 Pr = tan PR/2 



= tan (A - A)/2 = tan (\ + A)/2. 



The circle on the globe projects into a circle with centre on Psr, 

 passing through the points s and r. Hence its centre is at c, where 



„ 1 / X— A , A+A\ 



cos A + COS A 



and its radius will be 



»(' 



A + A . A — A\ Sin A 



J( tan —tan , 



2 2 / cos A + cos A 



The circle can thus be drawn after a very little computation, which 

 may be conducted either by use of 



tan (A + A)/2 and tan (A-A)/2, 

 or of the expressions 



sin A i sin A 



cos A + cos A ^ cos A + cos A 



In this way an epicentre can be very conveniently determined on a piece 

 of white paper. 



Sometimes the circle is very large and its centre may fall off the 

 paper in use. In this case it has been suggested by Mr. J. E. Pearson 



Fio. 2. 



(whose volunteer aid in thus determining epicentres is gratefully acknow- 

 ledged) that a very little numerical work will give the part of the circle 

 we want. Thus in Fig. 2 let N be the North Pole and let A and B 

 be the extremities of the diameter of the circle to be drawn. Let 

 NA = 6 inches and NB = 28 inches, so that B is quite off the paper, and 

 it is inconvenient to draw the circle. Nevertheless, we can quickly find 



