STUDIES IN STATISTICAL REPRESENTATION. 487 
that is to say, if the equation be applicable to the repre- 
sentation of a series of points 2, Y,............% Yx the 
logarithms of the coordinates will lie on a straight line. 
If A be not zero the logarithmic homologue is not a 
straight line. ‘To obtain the constants we must take three 
points on the curve the abscissae of which are in geo- 
metrical ratio: that is, we must obtain the values of y,, ¥2, 
y; for the abscissae x,, xk, «,k. 
Then we may write 
y=atber=at+L 
Y, = a + bak” = a + La, say, then 
Yo — @ + bok” — a +. La’. 
Consequently 
Ch = eC es 28 
i A L(a—1) oD 
that is, 
pee eg i Ya) WOR MG We) (29) 
log k 
When n is found, the constants a and bare readily deter- 
minable since x and k are also known. 
For mean values we may proceed analogously to the 
method indicated in Section 12 hereinafter. 
Equation (28) is unsuitable when the left hand member 
is negative. If not negative, then the curve is a parabola 
or hyperbola according as y; - y. is greater or less than yy, — 3. 
12. Exponential curves.—The general equation is 
de pee Wet (30) 
the logarithmic homologue of which, when A is 0, is 
logy = log B-+ na? log é.......:.--- (31) 
Hence if we take three ordinates whose abscissz are in 
geometrical progression x, xk, «wk? (k being known) the 
following equation can be deduced :— 
bie: ee (32) 
log y, — logy 
which determines p since k is known. 
