Some elementary properties of a function which is regular or meromorphic in a ring-region and has constant modulus on the boundary. For instance Theorem VI-let f(r)be meromorphic in 1<=|x|<=R, and|f(x)| be constant for |x| =1, =r, =R, 1<r<R, r2=R, then either f(x) must be Cxr, or. for rational s, |f(x)| must he constant for all |x|=R't/q, (t=1, 2, …q-1), s=q/p, and f(x) has at least one zero in every sub-ring-region and one pole in every complementary region. An application-Among all functions F(x) regular in r <|x| <R satisfying F(α)=α0, F'(α)=α1, …F(n)(α)=αn, there exists f(x) such that the upper limit of the moduli for r< x <R is smaller than that of any other F(x). f(x) is uniquely determined by α1α0, …αn, γ and R, and it is a rational function of g(x, α), of degree not exceeding n and of constant modulus on the boundary, where g(x, α) is a definite function which is regular in r<=x<=R and g(x, α)= 1 on the boundary and g(α, α)=0, g(x, α)≠α, g'(α, α)≠0. In Part I, I will give some elementary properties of a function which is regular or meromorphic in a circle, or in a ring-region and has constant modulus on the boundary of the region. Some of the properties are not new, but I will enumerate then in some order. Theorem VI will be perhaps new and of some interest. In Part II, I will give some theorems on the minimum of the maximum modulus of a functions which is regular in a ring-region under some given conditions as an application of the equi-modular functions

View full abstract

The tudumi a kind of drum with dumb-bell-shaped body. Present paper contains the results of investigations on the tone-colour of the two kinds of the udumi riz.the kotudumi and okawa.The sounds were recorded. similarly as in the previous works, by means of a condenser microphone and oscillograph. Generally speaking, the sound of the kotudumi contains, besides the fundamental, a partial tone having a frequeucy slightly smaller than thritce that of the fundamental. Among the four kinds of tone--ta, ti, pit and po produced by the kotudumi or shoulder drum, the pitch of the first two is in the neighbourhood of 3oo or d', while the other two are about one full tone lower. The po tone is the most characteristie, the tension of the membrane being removed immediately after a stroke, and it is found that the pitch is lowered about one full tone within 0.15 see. after the stroke. The sound of the okawa or sidedrum is found to be comparatively simple, consisting mainly of a tone having a pitch of about 650; the tone being therefore about one octave higher than the to and ti of the kotudumi. Its duration is very short, the amplitude decreasing to a half value within one fiftieth second

View full abstract

The motion of two vortices in a liquid was investigated photographically and it was clearly shown that there is a tendency to approach of two vortices with the same sense of rotation and to separate of those with the opposite senses

View full abstract