106 
MR. W. F. SHEPPARD ON THE APPLICATION OF THE THEORY OF 
For the standard form we take the curve whose semi-parameter is unity, and area 
unity. The central ordinate of this curve will for the present be denoted by C ; we 
shall show later that C = l/\/27r. It is clear that if A is the area of a curve of 
parameter 2a, its central ordinate is CA/a, 
The curve may he traced by means of Table I, (p. 153). The second column of that 
table gives the ordinate of the standard curve in terms of the abscissa; the third 
gives its ratio to the central ordinate. Table II. (p. 155) is formed by inverting 
this latter table ; it gives the abscissa in terms of the ratio of the ordinate to the 
central ordinate. 
§ 5. Moment-formulce .-—Let MP, M'P', be any two consecutive ordinates to a 
normal curve whose parameter is 2a, Draw Pvn and Vm perpendicular to the 
central ordinate OH, and let ]) and p be the intersections of MP, ?n'P' and of MT'. 
Fio” 2 
wiP respectively (fig, 2). Tlien, if PP' produced cuts the base in T, we have, by 
similar triangles, 
Pp'. MP = P'^/. MT = _pP, MT. 
Hence 
(1.) ()M X rectangle MP^;'M' = OM . Pp'. MP 
= OM . MT XpV = OM . MT (I\IP ~ M'P') ; 
(2.) OM“ X rectangle MP^:)'M' = OM . MT X nip ). pP 
= OM . MT X rectangle m'pVni ; 
(3.) OM^"^" X rectangle MPp'M' = OM. MT X wP* X rectangle m'pVm. 
The Z-’th moment of the rectangle nLp)Vm about OH is y—", . wP^ X nip)Vm. Also 
when MM' becomes indefinitely small, OM . MT = d\ Hence, by summation, we 
see that 
(i.) If MP and NQ are any two ordinates, the moment of the area MPQN 
about OH is a“ (MP — NQ); 
(iia.) If Pm and Qa are the perpendiculars from P and Q on OH, the second 
moment of MPQN about OH is d~ X area nQP7?i ; 
