446 
PROFESSOR K. PEARSOX OX .MATHEMATICAL 
The object of this supplement is to discuss the calculation of curves of Tvpe V.. 
and to consider tliose of Type VI. somewhat more at length, they being only hrieflv 
referred to on }). 369 of the memoir. It will be seen that Type I. of the memoir has 
now broken up into two divisions. One portion is the old Tyj)e I. passing into the 
normal curve on one side and Type III. on the other. This Type III. separates the 
second portion, Type VI., of the old Type I. from the first portion. Type VI. passes 
from Type III. to the new transition curve Type V., which, like Type III., will be 
found to Iiave a range limited in one direction only. Finally this new Type V. is the 
transition to the old Type IV. bounded on the other side by the sub-curve, the old 
Type II., and beyond that the normal curve. Thus we see that Types I. and IV. do 
not pass directly into each other through Type III., as might be supposed bv the 
criterion > or < 0, but that there are a series of intervening curves, two of which, 
Types V. and VI., require further consideration, if we are to complete the whole 
round of frequency distributions embraced under the differential equation (i.). 
(3.) On the Fre<iuenc>/ Curve of 'Type V. 
Returning to the fundamental differential equation (i.), let us consider what 
transformation takes ])lace when the denominator on the right has equal roots.* We 
may then write it in the form 
1 chf _ — X _1 1 
y dx Co (q -h xf ~ q (q -f- xf q (q + x) ' 
c I 1 
Hence log y = - - 4TX “ 7 (^1 + ^) + const. 
_v__ 
Thus y ■= y^c n +-‘' -f- x)~^, 
where, y^ is a constant, y = Cj/c^ and = I/cq. Thus changing the origin we may 
write the curve : 
y = 2/0 c"’"". 
where = yfp gives the distance of the mode from the new origin. 
To find the moments about this origin, we notice that, p and y being positive, 
y — 0 when X = 0 and when xx= cc . Thus as in the curve of Type III. we have a 
range limited at one end only. 
To find the moments we have, if a be the area, 
ay!„ = [ yQX~P"^" e~'^ ^ dx .(iv.). 
•' 0 
I owe to Miss Agnes Kelly, Ph.D., the suggestion that this type of frequency curve deserved fuller 
treatment. 
