/: 



156 Dr. J. BoTTOMLEV on 



In these equations <; is a variable parameter, and by giving 

 it successive values from until we exhaust the normals to 

 the first surface, we may obtain the equations to the 

 successive derived curves from the commencement of 

 dissolution until its completion. 



\i s and S denote the lengths of the derived and primi- 

 tive curves measured from two fixed points up to the 

 common normal, we may deduce from (3) 



ds _d^ (7) 



dx~dX' 

 and by integration 



VS d_x_ dX (8) 



dXdX 



from (4) by differentiation we obtain 

 dx . dd) 



// C T 



also -rf>= -: — ; substituting in (8) and completing the inte- 

 ^^r sm^ ' & V / r t> 



gration we obtain the equation 



s = S+C(j} + n, 

 n denoting a constant ; to find its value suppose that in 

 Fig. I MP = S and NO = j-, then we shall have simultane- 

 ously j = 0, S = 0, ^ = -; hence 

 equation may be written 



There is also another equation which may be deduced from 

 this, which will be found useful. Suppose that c is not 

 greater than the radius of curvature at any point of the 

 curve MK, and that OM, OK are normals, then the area 



MNLK may be written / sdc, if then we multiply (9) by 



dc, and integrate we get 



MNLK = &-^(;-,f) ('°' 



