The confidence limits (p1, p2) of occurrence probability p can be calulated form observed. relative frequenvy k/N by the following formulas. If we take ε=(100-2α)% as the confidence coeffcient, p1=n2/(n2+n1F1), where F1 is the α% point of F-distrbution with the degrees of freedom n1=2(N-k+1) and n2=2k, and p2=n1F2/(n2+n1F2), where F2 is the α% point of F-distribution with the degrees of freedom n1=2(k+1) and n2=2(N-k). Two kinds of tables and diagtrams are prepared, the one is for ε=90%(α=5%) and the other for ε=98%(α=1%). The table of (p1, p2) in which the arguments are N and k are used for 1_??_N_??_20, and the diagrams of (p1, p2) in which the arguments are N and k/N are used for 20_??_N_??_1000. Comparing with these tables and diagrams, the error of approximate confidence limits by binomial probability paper is negligible of N>50, N-5>k>5 or N>100, N>k>0. In the construction of an objective forcasting scheme, we can objectively proceed by making use of confidence limits of occurrence probability. Our tables and diagrams are also applied to find the confidence limits of corrlation coefficient calculated from the sings of variates. The diaram of arcsine transformation can be utilized for the purpose of testing the difference of two occurrence probabilities.
Considering the latent heat of water vapour, the vertical instability is; where T1' is the equivalent temperature obtained from the data of aerological observations, T2' the equivalent temperature when the previous stratum attains the state of equilibrium, and p0 the surface pressure. The author calculated this vertical instability at the time of the approach of tropical storms and investigated the distribution of the instability. It was found that there are greater vertical instabilities in the external region of storm and in the direction of the motion. In view of the above fact, we may conclude that Margules' vertical instability indicates the tendency of the energy release.
The amount of the precipition in forest and stem flow have been measured many times in the past but theoretical considerations of the relations between the amount of rainfall and the precipitation in forest or the stem flow seems to be lacking. The author tries to treat the problem theoretically and deduces some formulae. The author first makes clear the idea of the precipitaiton in forest. i. e he distinguishes between the part of precipitation that reaches the ground directly (including the drops of rain which touch the forest canopy but not caught by it and fall to the ground at once, probably splitting) and the other part which is caught by the leaves or twigs and then drips from them when their water retaining capacity exceeds a certain limit. He denotes the former by (1-α)re, re being the rainfall intensity and αre its fraction caught by the crown, and the latter by β(φ-φ0), φ being the amount of water retained by the crown, φ0 its threshold value for dripping, and β a proportional factor. He further denotes the stem flow by β'(φ-φ0'), where φ0 is a threshold value for stem flow and β' is another proportional facor. He then obtains the following equation for the time variation of the amount of water retained by the crown: (1) where γφ is the amount of evaporation assumed to be proportional to φ. Assuming α, β, β', γ, φ0 and φ0' are all constants, the equation gives a solution There are many experimental results of the precipitation in forest in which stem flow was not measured simultaneously, so it may be of value to duduce a formula for the total amount of precipitation that reaches the ground directly plus the total amount of dripping (=Ri'=Ri-S). This can be obtained at once from (16) and (18) as follows: (19) To see the behavier of the relations (16), (18) and (19) we can use the following approximations. If we assume Re is large or amall (and nearly equal to ret1), we can ezpand the second term on the right side of these equations and adopt the first two terms respectively. Then we have, The form of curves which are expected from these equations are shown in Figs. 3_??_7. Comparison with experiments may be done in many ways. Here the agreement of the equation (19) with the equation (24) was tested. αa3 and b3 in equation (24) were obtained from Japanese pecipitation measurements, where appropriate values were selected for α, φ0, t1 and β'/β. The agreement is good as is seen from Figs. 8 and 9. A similar comparison of (18) with the stem flow measurements by Kittredge, Laughead and Mazurak was done and was also found to be in good agreement as is seen in Figs. 12 and 13. There are, however, some uncertainties in the order of magntude of each constant, which will be the subject of a future article.