The positive sign is taken in [10] and the negative in [12] because r and r.^ 

 are taken intrinsically positive; but a is the slope of the sectional-area 

 curve at X = 0, and hence is positive, and a + 23^ + . . . + na is the slope 

 of the sectional-area curve at x = 1 , and hence is negative. If [11] is sat- 

 isfied, the body has a pointed tail and r^ = 0, so that [12] is valid for both 

 cases. 



The volume of the body may be expressed as 



1 1 



V = I TrY^dX = na^ I J y^dx, 







or, substituting for y^ from Equation [2], 



7^1^^.+ •■• +-HiT^n=¥% t^^l 



For convenience, the foregoing linear equations in the a 's are 

 assembled here: 



a^ + a^ + ... + a^ = [4] 



a^m + a^m^ + ... + a^m'^ = -jj- ^^j 



a^ + 2a2m + . . . + na^m""^ = [6] 



a, = 2r^ [10] 



a^ + 2a2 + . . . + na^ = -2r^ [12] 



2 ^ +i^2+ ••• +liT %=T% 



[13] 



SOLUTION OF EQUATIONS FOR POLYNOMIALS OF SIXTH DEGREE 



Corresponding to the parameters m^ , r , r^ , and C there are the six 

 equations [4], [5], [6], [10], [12], and [13]. Consequently a polynomial of 

 the sixth degree is, in general, determinable so that we choose n = 6 in these 

 equations. The solution of these equations by the determinant rule is tedious 

 and consequently an alternative procedure is developed. 



The form of the solution by the determinant rule shows that the a^^'s 



are linear functions of r , r, , and C . Hence y^ is also a linear function 



J- p " 



of r , ri , and C and may be written in the form 

 p ^ 



