MATHEMATICS: W. E. MILNE 
543 
system but is dependent upon many different factors which may some- 
times be opposing.^ Thus PauH and his pupils find minimum \dscosity 
at an isoelectric point in protein sols due to minimum ionization of the 
protein. In the writer's gelatine experiments, however, the gelation 
or gelation viscosity of the gel was distinctly at a maximum at the 
isoelectric point due to maximum aggregation. It appears necessary 
to distinguish between these two kinds of viscosity. 
Summary. — A close analogy to Osterhout's experiments on the elec- 
trical resistance of Laminaria is found in gelatine (plus NaOH), if we 
assume that the effect of time in the Laminaria experiments is to in- 
crease the concentrations of the salts in the cells of the tissue. 
1 These Proceedings, 2, 534 (1916). 
2 For the sake of brevity the word precipitate is used throughout to denote not only an 
actual precipitate, but any accompanying conditions which vary with the amount of pre- 
cipitate or the degree of precipitabiHty. 
3 Cf. Osterhout, Science, 41, 255 (1915) for summary of results. 
^Pauli has concluded for other reasons that protoplasm reacts much like protein soils 
containmg alkaK. Biochem. Zs., 24, 239 (1910). 
5 Samec, KoU.-Chem. Beihefle, 5, 141 (1913). 
^ Pauli, loc. cit. 
7 Osterhout, Science, 39, 544 (1914). 
8 Spaeth, Science, 43, 502 (1916). 
« Ostwald, Kolloid. Zs., 12, 213 (1913). 
ON CERTAIN ASYMPTOTIC EXPRESSIONS IN THE THEORY 
OF LINEAR DIFFERENTIAL EQUATIONS 
By W. E. Milne 
DEPARTMENT OF MATHEMATICS. BOWDOIN COLLEGE 
Received by the Academy, July 6. 1916 
The nature of the solutions of a certain linear differential equation 
containing a complex parameter has been investigated by Prof. G. D. 
Birkhoff,^ who discovered the asymptotic character of the solutions 
when the parameter is large in absolute value. These results he em- 
ployed in the study of expansion problems connected with the particular 
differential equation 
^ + i'2W^+ . . . +P„W« + p"« = o, (1) 
together with n linearly independent linear boundary conditions 
W,{u)=0,W2{u)=0, . . . ,WAu)=0, (2) 
It is the aim of this paper to present asymptotic formulas for n linearly 
independent solutions yi, y^, . . . , y,, of equation (1) which are in 
