36 



C. V. L. Charlier. 



known quantity are existing, is, however, a very laborious operation. Mr Pearson 

 has indeed possessed the energy to perform this heroic task in some instances 

 in his first memoir on tliese topics from tlie year 1894. But I fear that he will 

 have few successors, if the dissection of the frequency curve into two components 

 is not very urgent. 



A somewliat less tedious work may lead to the knowledge of the roots, if we 

 start from the two equations (41). 

 Writing 



( = 6C,n-'' — 3C.«- — eC^Ci« — öC«, 



(45) I tr, = 2.r'+3C,« + 4Ci, ' 

 I 2U., = im' + 3'C,u + 3C|, 



we have 



(46) ; u, 



and here f/j , U.^ and U.^ are polynoms in n of the third degree. If the roots of 

 the equations î/^ = U.2 = U.^ = 0 be known, the roots of the nonic may be easily 

 discussed without solving the equation (42). 



With this aim we construct the two curves dehned by (40). We call them I 

 and II. If 



= (u — {u — «2) [u — fly), 



we find that I has infinite branches for u = b^, u = h.^ and ti = h^. The curve II 

 has generally a parabola-like appearance. Supposing and to be imaginary we 

 have for instance the following form of the curves I and II — Oj, a^, and h^, 

 h^, being supposed to be all real. 



We find from inspection that we must possess in this case 5 real roots of the 

 nonic, the approximate values of which are directly found from the figure. For a 

 more detailed knowledge of the roots we may calculate the curves more accurately 

 in the neighbourhood of these approximate values. 



I have applied this method to some instances and have found the determi- 

 nation of the values of the roots in this mannef tolerably easy. 



There is, however, enough labour left to discourage an inquirer from ope- 

 rating an mathematical dissection of a given frequency curve. In some instances 

 the operation may be performed in an easier manner. 



l:o Suppose the values of and h.^ to he given. The dissection of the frequency 

 curve is then very easy. Using the same denominations as before {l>^=^ y^, ^2 = */g 

 a. s. o.) we get and fi'om the relations 



(.'/i — .'/a) -"2 = Vi 



