1. We have investigated the theoredtical and practival problems associated with the interpreataion of time constants and the estimation of electrotonic length with equaivalent cylinder formulas for neurons best respresented as amultiple cylinders or branched structures. Two analytic methods were used to compute the time constants and coefficients of passive voltage transients (and time constans of current transient under voltage clamp). One method, suitable for simple geometries, involves analytic solutions to boundary value problems. The other, suitable for neurons of any geometric complexity, is an algebraic approach based on compartmental models. Neither of theses methods requires the simulation of transients. 2. We computed the time constants and coefficients of voltage transients for several hypothetical neurons and also for a spinal motoneuron whose morphology was characterized frrom serial reconstructions. These time constants and coefficients were used to generate voltage transients. The exponential peeling, nonlinear regression, and transfrom methods, were applied to these transients to test how well these procedures estimate the underlying time constants and coefficients. 3. For a serially reconstructed motoneuron with 732 compartments, we found that the theoretical and peeled tau0 values were nearly equal, but the theoretical tau1 was much larger than the peeled tau1. The theoretical tau1 could not be peeled because it was associated with a coefficient, C1, that has a very small value. In fact, ther were 156 time constants between 1.0 and 6.0 ms, most of which had very small coefficients; none had a coefficient larger than 2% of the signal. The peeled value of tau1 (called tau1 peel) can be viewed as asome sort of a weighted average of the time constants having the largest coefficients. 4. We studied simple hypothetical neurons to determine what interpretation could be applied to the multitude of theoretical time constants. We found that after tau0, there was a group of time constants associated with eigenfunctions that were odd(or approximately odd) functions with respect to the soma. These time constants could be interpreted as "equalizing" time constants along particular paths between different pairs of dendritic terminals in the neuron. After this group of time constants, there was one that we call tau(even) because it was associated with an eigenfunction that was approximately even with resperct to the soma. This tau(even) could be interpreted as an equalizing time constant for charge equalization between proximal membrane (soma and proximal dendrites) and distal membrane (including all distal dendrites). 5. The peeled time constant tau1 peel will equal tau(even) whenever the neuronal morphology can be reduced toa an equaivalent cylinder. In these cases the time constants associated with odd eigenfunctions have zero coefficients. However, when the neuronal morphology does not satisfy the constraints for reduction to a cylinder, the coefficients of the time constants associated with approximately odd eigenfunctions are usually small but not zero (for input and recording at athe some). The effect of these coefficients not being zero is to interfere with getting a tau1 peel equal to tau(even). When the equaivalen cylinder formul, L(tau0/tau1) = pi/(tau0/tau1 - 1)1/2, is applied to nonequivalent cylinder neurons, the hope is that tau1 peel is approximately equal to tau(even). Electrotonic length estimates that use tau1 peel for tau1 in this formula will be in error depending on how different tau1 peel is from tau(even). 6. Electrotonic length estimates that make use of tau1 are highly sensitive to experimetnal noise an problems inherent in the proceudres used to estimate exponentials from data. We compared exponential peeling, nonlinear regression, and transform methods (DISCRETE) as methods for resolving exponentials from noisy and noise-free voltage transients. Nonlinear regression and DISCRETE estimated the actual tau1 more accuarately than exponential peelin when applied to noise-free transients. However, with noisy transients, the results with all three method were similar; the tau1 estimates were almost always closer to tau(even) that to the actual tau1, and consequently, the L estimates were usually close to the average terminal L value. 7. In conclusion, the tau1 peel that one estimates from experimental data may be significantly different from the actual tau1 of the neuron. The error in L estimates computed with tau1 peel depends on how different tau1 peel is from tau(even). when membrane resistivity is uniform and the dendrites end at electrotonic distances within 10-2-% of each other, tau1 peel will be close to tau(even), and L(tau0/tau1) will be close to the average L (weighted by input conductance). If the dendrites end at very diffeent electrotonic distances, tau1 peel may be much larger tha tau(even), and L(tau0/tau1) may be much larger than the average L; with a noise-free transient, the L estimate may even approach the longest tip-to-tip electrtonic distance path in the neuron. However, noise and problems inherent in estimating exponentials from data have very much limited the magnitude of over-estimates (due purley to violatins of the morphological assumptions of the equivalent cylinder model) in L values reported to date. Methods for estimateing electrotonic length in nonequivalent cylinder neurons that avoid the problems of the interpretation and estimation of tau1 and that also take into account a possible soma shunt are presented in the next two papers.