160 MEMOIRS OF THE NATIONAL ACADEMY OF SCIENCES. 



represent the normal wave-length scale, and upon a line, CD, at right angles to the first, lay off 

 the same distance and divide it into the same number of parts, spaced according to the law of 

 dispersion of the prism, as iu the wave-length scale marked on the bottom of tlie prismatic chart, 

 Plate I. Erect ordinates at the points of division, and mark them with the i)roper wave-lengths, 

 beginning on both lines at the ends which lie nearest to each other, as in the figure, where five 

 ordinates are shown ; through the intersection of corresponding ordinates draw the curve EF and 

 upon CD draw the curve of distribution of energy in the piismatic spectrum. 



Let a, Fig. 4, be a very small wave-length interval on the iirismatic scale; c, the same interval 

 on the normal scale, and 6 and d the average heights of the energy curves over the two intervals, 

 respectively ; the shaded part of the figure representing, therefore, the portion of the total area 

 included between these limits, e/" is a portion of the curve EF, Fig. 3. Then, according to the con- 

 dition of transformation, 



cd = ab 

 whence 



b: d:: c: a 

 From geometrical considerations, 



c : a:: 1 : tan tp 



where ip is the angle which the cord EF, joining the intersections of the two pairs of ordinates 

 makes with AB ; consequently 



b: d:: 1 : tan cp 

 from' which 



d = b tan (p 



Now, when a and c are indefinitely small, b and d are the ordinates of the prismatic and 

 normal energy curves, respectively, at a given wave length, and q) is the angle formed by the 

 tangent to EF at their point of intersection. Hence, to find the height of the normal curve at a 

 given wave-length, the corresponding ordinate of the prismatic curve must be multiplied by tan cp. 



Such a construction was applied to the prismatic energy curve of the Hilger prism. 



The true normal energy curve with all its inflections, maxima and minima, is easily drawn 

 after this (dotted) bounding curve of normal energy is plotted, for the parts of the ordinate of the 

 latter below and above its intersection with the Ibrmer irregular curve bear the same proportion 

 to each other as in the prismatic spectrum, and we thus finally attain the object of tlie preceding 

 labor. 



If, now, it is desirable to map the distribution of the energy on any other senile, such as that 

 on which the abscissfe are proportional to the times of vibration, this can be done with facility. 

 Thus, in the supposed instance, we have only to find corresponding to each wave-length in order 



to get the abscissa:', and (observing that since a; now =-,^^ =— ) to use the multiplying factor 



to obtain the length of the new abscissae from the old in each instance. If the length of the new 

 energy curves between the limiting perpendiculars (which now represent the reciprocals of the 

 wave-length), is to be the same as in the old, we nmst introduce a constant multii)lier, «, writing 



the equation of the interpolating curve .v— j, so that the multiplying factor becomes — ^^. 



Thus if the limiting ordinates of the wave-length energy curve are /!„, ^,,, and we are to have the 



condition f — A h = A,, — A,„ n = A„ x Aj, &c. 



If the mean ordinate of any small area of the normal energy curve between any given limits, 

 A„, A„ is denoted by 2/,, and that of the corresponding area of the new curve by y, since the areas 



