490 On an Apparatus for integrating Differential Equations. 



similar, and the lines bb, cc make angles of 45° with the ver- 

 tical, P describes a series of concentric circles. 



(2) Similar to (1) except that the template G is inverted, 

 and the lines bb, cc slope in opposite directions. 



Now <j>(q)=mq i ty{r) = k—nr, f l (t)=bt. f,(t) = -ct. 



The differential equation becomes 



(m — n) xdx- (mb + nc) (xdt + tdw) -f (mB—nc*)tdt + (k — M)r7#=0. 



P describes a series of similar and coaxial hyperbolas. 



(3) An harmonic synthesizer or tide predicter may be con- 

 structed of a number of rectangular or cylindrical plungers 

 of sectional areas corresponding to the amplitudes of the several 

 components, lifted and lowered in the tank by eccentrics of 

 the proper periods. The level of the water will give the 

 synthesis required. 



(4) Suppose that in Petrovitch's instrument the curve 

 described by the pointer attached to the moving template is 

 a straight line y = t, and that the curve drawn by the pencil 

 which follows the water-level is the curve x = <j> (t), so that 

 #=<£(y). Now cause the pointer to follow the curve 



y=- sin" t, so that the new curve drawn by the pencil is 



#=<£>(- sin" t\ 



The area of the last curve is 



U(isin- ] £W= U(0) cos nddO, 



where £=sin n6. Thus the Fourier components of the 

 curve x = <f>{i) can be obtained if the form of the templates 

 can be discovered which will develop the curve itself from a 

 straight-line course for the pointer. 



A great number of forms in which instruments can be made 

 on the same general principle will readily suggest themselves. 

 One consists of several vessels of different shapes containing 

 water which may be transferred between the vessels by means 

 of taps, or by a ladle. If the heights of the water-levels in 

 the several vessels are x,y,z, . . . and the areas of the water- 

 surfaces X, Y, Z, . . . ; then Xdx + Ydy + Zdz+ . . . = and 

 curves or surfaces drawn by plotting the values of x, y, z, or 

 by a mechanism attached to floats give solutions of the 

 equation. 



