BOCHER. — A PROBLEM IN STATICS. 483 



and this is the equation for the functions of the parabolic cylinder in the 

 form most commonly used. That in the special case we have just been 

 considering (the case in which ^ is a positive integer) this equation has a 

 solution in the form of the product of an exponential function and a poly- 

 nomial was noticed by K. Baer.* I am not aware, however, that the 

 connection between this fact and Hermite's work on the one hand and 

 Stieltjes's method on the other has ever been pointed out. 



The field of force K{ — c x) which we have just considered may be 

 regarded as the limit of the field due to two particles of mass m situated 

 at the points e and — e and two particles of mass — m situated at i e and 

 — ze, each particle repelling directly as the mass and inversely as the 

 distance, as e and m both become infinite. That is, the field may be 

 regarded as due to a quadruplet at infinity. A simpler case, from this 

 point of view, would be that of a doublet at infinity which would produce 



a uniform field, let us say Ki — - J. Here, however, there is clearly 



no position of equilibrium. In order to make equilibrium possible, let 

 us introduce into this uniform field a single fixed particle of mass m, 

 repelling inversely as the distance, and situated, say, at the origin. This 

 problem can be carried through precisely as was the one above, and leads 

 us, not to equation (5), but to: 



d"^- dz 



(6) 4a;^J, + 8 m — = \c'' x _ 4c (m + k)]z, 



and we see again that if h is a positive integer, (6) has a solution which 

 is the product of an exponential factor and a polynomial, the roots of 

 this polynomial being the positions of equilibrium in the problem last 

 considered. 



It is interesting to note that equation (6), in the special case m = ^, 

 reduces to the form in which the equation for the functions of the para- 

 bolic cylinder presents itself when the subject is approached from a 

 broader point of view than is ordinarily done.f 



We may, of course, consider doublets, triplets, etc., which do not lie 

 at infinity, and such centres of force may come in in any number along- 



* Programm, Kiistrin, 1883, p. 9. 



t Cf. the book of tlie present writer already referred to. By introducing x- in 

 place of X as the independent variable in (5), and 2 k in place of k, we can pass at 

 once to the special case of equation (6) in which m = i. 



